## Undergraduate Honors Theses

#### Thesis Defended

Spring 2019

#### Document Type

Thesis

#### Type of Thesis

Departmental Honors

#### Department

Mathematics

#### First Advisor

Katherine Stange

#### Abstract

Let $f(x) \in \ZZ[x]$ be a monic, irreducible polynomial and $\theta$ a root of $f$. Further, we define the field extension $K := \QQ(\theta)$ and denote its ring of integers by $\mathcal{O}_K$; an essential task in Algebraic Number Theory is to compute the factorization of ideals of $\mathcal{O}_K$ into prime ideals. We begin this thesis by reviewing some basic properties of number fields. Then, we describe a classical result by Dedekind which gives the factorization of the ideal $p\mathcal{O}_K$ for all primes $p$ not dividing the index $\left[\mathcal{O}_K:\ZZ[\theta]\right]$. Finally, we discuss the Montes algorithm which builds off the previous work of Ore in exploiting certain Newton polygons to encode the data needed to create a general factorization algorithm.

#### Recommended Citation

Ibarra, Ryan, "Factorization of Ideals in Algebraic Number Theory and the Montes Algorithm" (2019). *Undergraduate Honors Theses*. 1792.

https://scholar.colorado.edu/honr_theses/1792